With iOS 15 and iPadOS 15, you can change your name and photo in Messages. Here’s how:
° Launch Messages.
° Tap Edit in the upper-left corner.
° Tap Edit Name and Photo.
Edit your name. Or tap the image circle to choose an image, photo, emoji, or Memoji. (Memoji is the name used for iOS’s personalized “Animoji” characters that can be created and customized right within Messages by choosing from a set of inclusive and diverse characteristics to form a unique personality.)
° Tap Done.
Your name and photo can be shared with anyone using iMessage, but you’ll be asked to confirm that you want to share with someone when sending or replying to a new message. You can also turn sharing on and off. Just complete steps 1-2, and turn Name and Photo Sharing on or off.
(This how-to is based on my experiences and info on Apple’s support pages — where the images sometimes come from.)